Answer to Question #25248 in Abstract Algebra for Mohammad

Question

Show that the following are equivalent:
(A) If I is a nil ideal in any ring R, then Mn(I) is nil for any n.
(B)' If I is a nil ideal in any ring, then M2(I) is nil.

Answer 1

That (A) implies (B) is obvious,since it is case n=2.For, if (B) holds, then for any nilideal I, m=2^t , M_m (I) is isomorphic to M₂(M_m/2(I)) is also nil, by induction on t. Since any n is bounded by 2^t forsome t, we have (A).